Theorem onintexmid 4720

Description: If the intersection (infimum) of an inhabited class of ordinal numbers belongs to the class, excluded middle follows. The hypothesis would be provable given excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Aug-2021.)

Hypothesis

Ref	Expression
onintexmid.onint	⊢ ((𝑦 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝑦) → ∩ 𝑦 ∈ 𝑦)

Assertion

Ref	Expression
onintexmid	⊢ (𝜑 ∨ ¬ 𝜑)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)

Proof of Theorem onintexmid

Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prssi 3873	. . . . . 6 ⊢ ((𝑢 ∈ On ∧ 𝑣 ∈ On) → {𝑢, 𝑣} ⊆ On)
2		prmg 3835	. . . . . . 7 ⊢ (𝑢 ∈ On → ∃𝑥 𝑥 ∈ {𝑢, 𝑣})
3	2	adantr 276	. . . . . 6 ⊢ ((𝑢 ∈ On ∧ 𝑣 ∈ On) → ∃𝑥 𝑥 ∈ {𝑢, 𝑣})
4		zfpair2 4347	. . . . . . 7 ⊢ {𝑢, 𝑣} ∈ V
5		sseq1 3271	. . . . . . . . 9 ⊢ (𝑦 = {𝑢, 𝑣} → (𝑦 ⊆ On ↔ {𝑢, 𝑣} ⊆ On))
6		eleq2 2302	. . . . . . . . . 10 ⊢ (𝑦 = {𝑢, 𝑣} → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ {𝑢, 𝑣}))
7	6	exbidv 1878	. . . . . . . . 9 ⊢ (𝑦 = {𝑢, 𝑣} → (∃𝑥 𝑥 ∈ 𝑦 ↔ ∃𝑥 𝑥 ∈ {𝑢, 𝑣}))
8	5, 7	anbi12d 477	. . . . . . . 8 ⊢ (𝑦 = {𝑢, 𝑣} → ((𝑦 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝑦) ↔ ({𝑢, 𝑣} ⊆ On ∧ ∃𝑥 𝑥 ∈ {𝑢, 𝑣})))
9		inteq 3973	. . . . . . . . 9 ⊢ (𝑦 = {𝑢, 𝑣} → ∩ 𝑦 = ∩ {𝑢, 𝑣})
10		id 19	. . . . . . . . 9 ⊢ (𝑦 = {𝑢, 𝑣} → 𝑦 = {𝑢, 𝑣})
11	9, 10	eleq12d 2309	. . . . . . . 8 ⊢ (𝑦 = {𝑢, 𝑣} → (∩ 𝑦 ∈ 𝑦 ↔ ∩ {𝑢, 𝑣} ∈ {𝑢, 𝑣}))
12	8, 11	imbi12d 234	. . . . . . 7 ⊢ (𝑦 = {𝑢, 𝑣} → (((𝑦 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝑦) → ∩ 𝑦 ∈ 𝑦) ↔ (({𝑢, 𝑣} ⊆ On ∧ ∃𝑥 𝑥 ∈ {𝑢, 𝑣}) → ∩ {𝑢, 𝑣} ∈ {𝑢, 𝑣})))
13		onintexmid.onint	. . . . . . 7 ⊢ ((𝑦 ⊆ On ∧ ∃𝑥 𝑥 ∈ 𝑦) → ∩ 𝑦 ∈ 𝑦)
14	4, 12, 13	vtocl 2877	. . . . . 6 ⊢ (({𝑢, 𝑣} ⊆ On ∧ ∃𝑥 𝑥 ∈ {𝑢, 𝑣}) → ∩ {𝑢, 𝑣} ∈ {𝑢, 𝑣})
15	1, 3, 14	syl2anc 415	. . . . 5 ⊢ ((𝑢 ∈ On ∧ 𝑣 ∈ On) → ∩ {𝑢, 𝑣} ∈ {𝑢, 𝑣})
16		elpri 3732	. . . . 5 ⊢ (∩ {𝑢, 𝑣} ∈ {𝑢, 𝑣} → (∩ {𝑢, 𝑣} = 𝑢 ∨ ∩ {𝑢, 𝑣} = 𝑣))
17	15, 16	syl 14	. . . 4 ⊢ ((𝑢 ∈ On ∧ 𝑣 ∈ On) → (∩ {𝑢, 𝑣} = 𝑢 ∨ ∩ {𝑢, 𝑣} = 𝑣))
18		incom 3421	. . . . . . 7 ⊢ (𝑣 ∩ 𝑢) = (𝑢 ∩ 𝑣)
19	18	eqeq1i 2246	. . . . . 6 ⊢ ((𝑣 ∩ 𝑢) = 𝑢 ↔ (𝑢 ∩ 𝑣) = 𝑢)
20		dfss1 3435	. . . . . 6 ⊢ (𝑢 ⊆ 𝑣 ↔ (𝑣 ∩ 𝑢) = 𝑢)
21		vex 2824	. . . . . . . 8 ⊢ 𝑢 ∈ V
22		vex 2824	. . . . . . . 8 ⊢ 𝑣 ∈ V
23	21, 22	intpr 4002	. . . . . . 7 ⊢ ∩ {𝑢, 𝑣} = (𝑢 ∩ 𝑣)
24	23	eqeq1i 2246	. . . . . 6 ⊢ (∩ {𝑢, 𝑣} = 𝑢 ↔ (𝑢 ∩ 𝑣) = 𝑢)
25	19, 20, 24	3bitr4ri 213	. . . . 5 ⊢ (∩ {𝑢, 𝑣} = 𝑢 ↔ 𝑢 ⊆ 𝑣)
26	23	eqeq1i 2246	. . . . . 6 ⊢ (∩ {𝑢, 𝑣} = 𝑣 ↔ (𝑢 ∩ 𝑣) = 𝑣)
27		dfss1 3435	. . . . . 6 ⊢ (𝑣 ⊆ 𝑢 ↔ (𝑢 ∩ 𝑣) = 𝑣)
28	26, 27	bitr4i 187	. . . . 5 ⊢ (∩ {𝑢, 𝑣} = 𝑣 ↔ 𝑣 ⊆ 𝑢)
29	25, 28	orbi12i 776	. . . 4 ⊢ ((∩ {𝑢, 𝑣} = 𝑢 ∨ ∩ {𝑢, 𝑣} = 𝑣) ↔ (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢))
30	17, 29	sylib 122	. . 3 ⊢ ((𝑢 ∈ On ∧ 𝑣 ∈ On) → (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢))
31	30	rgen2a 2604	. 2 ⊢ ∀𝑢 ∈ On ∀𝑣 ∈ On (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢)
32	31	ordtri2or2exmid 4718	1 ⊢ (𝜑 ∨ ¬ 𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 720   = wceq 1402  ∃wex 1545   ∈ wcel 2209   ∩ cin 3219   ⊆ wss 3220  {cpr 3710  ∩ cint 3970  Oncon0 4508

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-uni 3936  df-int 3971  df-tr 4230  df-iord 4511  df-on 4513  df-suc 4516

This theorem is used by: (None)